Golden Germ Fourth-Order Ledger

Abstract

A finite signed fourth-order correction cancels the two surviving local modes below beta six and leaves a norm-summable prime deviation.

Theorem 1.1 (Fourth-order local cancellation reaches the beta-six boundary).

$$\begin{array}{r}\forall s\in \mathbb{C},\\ \frac{1}{o5Beta(6)}<\Re (s)\Rightarrow \\ x(p):=p^{-s\times {\phi }^2},y(p):=p^{-s\times {\phi }^3},\\ T7(p):=\sum _{k\ge 7}{p}^{-s\times o5Beta(k)},\\ K3(p):=(1-y(p{)}^2{)}^{-1}\times (1-x(p{)}^2\times y(p))\times (1-y(p))\times (1+x(p){)}^{-1}\times \mathrm{germLocalFactor}(s,p),\\ C4(p):=(1-x(p)\times y(p{)}^2)\times (1-x(p{)}^3\times y(p){)}^{-1},\\ H4(p):=-x(p{)}^5\times y(p{)}^6+x(p{)}^5\times y(p{)}^4-x(p{)}^4\times y(p{)}^6+x(p{)}^4\times y(p{)}^4-x(p{)}^4\times y(p{)}^2+x(p{)}^4\times y(p{)}^1+x(p{)}^3\times y(p{)}^4-x(p{)}^3\times y(p{)}^3+x(p{)}^2\times y(p{)}^5-x(p{)}^2\times y(p{)}^2+x(p{)}^1\times y(p{)}^4-x(p{)}^1\times y(p{)}^3,\\ R4(p):=(1-x(p{)}^3\times y(p){)}^{-1}\times (1-y(p{)}^2{)}^{-1}\times (1+x(p){)}^{-1}\times (H4(p)+(1-x(p)\times y(p{)}^2)\times (1-x(p{)}^2\times y(p))\times (1-y(p))\times T7(p)),\\ (\frac{1}{{\phi }^5}<\Re (s)\Rightarrow \mathrm{Summable}(p:\mathrm{Primes}(\mathbb{N})\mapsto |K3(p)-1|))\wedge \\ (1\times {\phi }^2+2\times {\phi }^3<o5Beta(6)\wedge 3\times {\phi }^2+1\times {\phi }^3<o5Beta(6))\wedge \\ (\forall a,b\in \mathbb{N},(a,b)\in \{(5,6),(5,4),(4,6),(4,4),(4,2),(4,1),(3,4),(3,3),(2,5),(2,2),(1,4),(1,3)\}\Rightarrow o5Beta(6)\le a\times {\phi }^2+b\times {\phi }^3)\wedge \\ (\forall k,i,j,l,m,n,r\in \mathbb{N},o5Beta(6)\le o5Beta(k+7)+i\times ({\phi }^2+2\times {\phi }^3)+j\times (2\times {\phi }^2+{\phi }^3)+l\times {\phi }^3+m\times (3\times {\phi }^2+{\phi }^3)+n\times (2\times {\phi }^3)+r\times {\phi }^2)\wedge \\ (\forall p\in \mathrm{Primes}(\mathbb{N}),C4(p)\times K3(p)=1+R4(p))\wedge \\ \mathrm{Summable}(p:\mathrm{Primes}(\mathbb{N})\mapsto |C4(p)\times K3(p)-1|)\wedge \\ \frac{1}{o5Beta(6)}<\frac{1}{10}.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/EulerGerm/GoldenGermFourthOrderLedger.golden_fourth_normalized_factor_deviation_norm_summable (✓ std3). ∎

Source. Repository-derived.

Commentary.

The frozen fourth-order exponent census supplies only the candidate weights below beta six; it does not determine which modes occur or cancel. This module’s seven-mode local expansion and the identity fourth_local_identity determine x y-squared and x-cubed y as the actual surviving correction modes. The signed factor C4 is therefore one minus x y-squared times the inverse of one minus x-cubed y. The displayed twelve-term polynomial H4 has no monomial below beta six; x-squared y-squared is its boundary term.

The theorem reuses the frozen third-order local factor K3 and records the exact rational identity C4 K3 equals one plus R4. Its shifted seventh-mode tail starts above beta six, and every factor arising from the numerator and denominator expansions adds a nonnegative mixed weight. Prime rpow summability and uniform geometric denominator bounds then prove norm summability whenever the real part of s is greater than one over beta six.

This is the next finite certificate in the golden Euler germ extraction ladder of OACTC parts 580 and 581. It advances the open local summability boundary from one over phi to the fifth power to one over beta six. It does not assert O-5, the Riemann Hypothesis, a global continuation or nonvanishing theorem, or an all-order extraction.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermFourthOrderLedger.golden_fourth_normalized_factor_deviation_norm_summable
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermFourthOrderExponentCensus
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermThirdOrderFactorization
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermThirdOrderLedger